Hamiltonian systems, symplectic flows, classical integrable systems

**3**

votes

**2**answers

238 views

### Almost Toric Symplectic Four-Manifolds

Let $(M,\omega)$ be a closed, symplectic four-manifold admitting an almost toric fibration, in the sense of Symington and Leung (e.g. http://arxiv.org/pdf/math/0210033.pdf). That is, there is a ...

**4**

votes

**1**answer

127 views

### use Floer homology to prove the fixed points

I read paper, in page 21, there is a proposition:
Let $(M,\omega)$ be a closed symplectic manifold with $\pi_2(M)=0$. Let {$f_t$}, $f_0$ = id, $f_1= f$ be a Hamiltonian path on M generated by a ...

**1**

vote

**0**answers

50 views

### About interpolability of Stein structures

Imagine $V$ is a complex vector space and $U_1\subset U_2\subset V$ two Euclidean balls.
Let $\psi,\psi_1$ be strictly plurisubharmonic functions on $V$ and $U_1$ respectively.
Question: What are ...

**3**

votes

**1**answer

144 views

### All symplectic manifolds have $Mp^c$-structures?

Let $(M,\omega)$ be a symplectic manifold, and $Mp^c(n)=Mp(n)\times_{\mathbb Z_2}U(1)$ which $Mp(n)$ here is Metaplectic group which is the double cover of symplectic group. I am looking for a ...

**2**

votes

**0**answers

194 views

### Generalizing a result of Paul Andi Nagy

I am trying to generalizate a result of Paul Andi Nagy which says that in a almost Kähler manifold with parallel torsion we have $\langle \rho,\Phi-\Psi\rangle $ is a nonnegative number; in fact
$$
...

**1**

vote

**2**answers

121 views

### Chern classes of reduced space for Hamiltonian circle action

I have a question about Chern class of symplectic reduction.
Let $(M,\omega)$ be a smooth compact symplectic manifold with a Hamiltonian circle action.
Let $H : M \rightarrow \mathbb{R}$ be the ...

**1**

vote

**1**answer

152 views

### Relation between volume of reduced space and phase space

Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if $J:M\to \frak g^*$ be its equivariant moment map then the reduced space is ...

**4**

votes

**2**answers

892 views

### A question about Marsden-Weinstein reduction theory

Let $G$ be a compact Lie group and $\frak g$ be its Lie algebra. Then by Marsden-Weinstein reduction theory we know that if we take $M=T^*G$ and $J \colon M\to \frak g^*$ be its moment map then the ...

**0**

votes

**1**answer

233 views

### Relation between kahler potential and Hermitian metric

Let $(M,\omega)$ be a Kaehler manifold and $h$ be its Hermitian form, then in local sense we can write $$\omega=\partial\bar\partial\log h$$ and also if $f$ be the kaehler potential then we can write ...

**4**

votes

**0**answers

107 views

### Concrete almost-complex structures on $3 \#CP^2$

The connect sum $X:=CP^2\# CP^2 \# CP^2$ supposedly supports almost-complex structures, i.e. endomorphisms $J$ of the tangent bundle such that $J^2=-id$. The existence of these almost-complex ...

**3**

votes

**0**answers

90 views

### DGBV algebra of symplectic manifold

Let $(M,\omega)$ be a simply connected closed symplectic manifold. Then we have the symplectic codifferential operator $d^{\star}$. Furthermore, $(\Omega^{*}(M),d,d^{\star})$ is a differential ...

**3**

votes

**0**answers

118 views

### Moment map in the singular case

The moment map is defined on the symplectic manifold $(M,\omega)$, or particularly, $(M,\omega)$ is Kahler. While $\omega$ is smooth or differential enough, the definition is obvious to understand, in ...

**1**

vote

**1**answer

46 views

### Reduction along an Orbit for C.-M. systems

I am having trouble in understanding the section of this paper http://www-math.mit.edu/~etingof/zlecnew.pdf
where the author introduces the Calogero-Moser system as the reduction of a manifold $M$ on ...

**2**

votes

**1**answer

290 views

### Cotangent bundle of coadjoint orbit is stein manifold?

Let me first define stein manifolds and coadjoint orbits.
A complex manifold $X$ of complex dimension $n$ is called a Stein manifold if the following conditions hold:
$X$ is holomorphically convex, ...

**7**

votes

**1**answer

318 views

### When a symplectic manifold is formal?

Let $(M,\omega)$ be a $2n$-dimensional symplectic manifold, then we have the symplectic Hodge operator
$$*:\Omega^{k}(M)\rightarrow\Omega^{2n-k}(M)$$
Furthermore, we can define a differential ...

**0**

votes

**0**answers

161 views

### Symplectic submanifolds in $\mathbb{R}^4$

Which symplectic submanifolds can be realized in $\mathbb{R}^4$ with standard ($\text{d}\,\boldsymbol{p} \wedge \text{d}\,\boldsymbol{q}$) symplectic structure? It's easy to show that such ...

**2**

votes

**1**answer

190 views

### Extending Reeb field from contact submanifold to ambient contact manifold

Let $(Y,\lambda)$ be a contact manifold, with a codimension-2 contact submanifold $(S,\lambda|_S)$ (this requires $TS\pitchfork\text{Ker}\lambda$). On $Y$ there is a natural vector field, the Reeb ...

**2**

votes

**1**answer

149 views

### Anti_symplectic 2-forms

A two form $\alpha$ on a n- manifold $M$ is called anti symplectic if for every $x\in M$, $\{ v\in T_{x} M \mid i_{v} \alpha=0 \}$ is a $n-2$ dimensional subspace of $T_{x}M$. So we obtain a $n-2$ ...

**2**

votes

**0**answers

83 views

### Is it true that a nondegenerate minimizing periodic orbit of mechanical Hamiltonian system is hyperbolic

Consider mechanical Hamiltonian system of the form
$$H(p,q)=\dfrac{\Vert p\Vert^2}{2}+V(q),\quad (q,p)\in T^*\mathbb T^n.$$
Here we suppose the periodic orbit $\gamma$ minimizes the Lagrangian ...

**3**

votes

**0**answers

224 views

### Transversality in Bourgeois Oancea's non-equivariant contact homology

In their paper, "AN EXACT SEQUENCE FOR CONTACT- AND SYMPLECTIC HOMOLOGY", Bourgeois and Oancea defined, whenever there is sufficient transversality, a "non-equivariant contact homology". Essentially, ...

**29**

votes

**1**answer

816 views

### What is the meaning of $(h^{11},h^{21})\to (h^{11}-240,h^{21}+240)$ in Calabi-Yau threefolds?

By browsing through the Hodge data of known Calabi-Yau threefolds, I stumbled upon an observation that frequently enough a pair of Hodge numbers $(h^{11},h^{21})$ comes together with the pair $ ...

**2**

votes

**1**answer

120 views

### Computation of symplectic quasi-state

A subset of a symplectic manifold is called strongly non-displaceable if it cannot be displaced by symplectomorphisms. A meridian in a $2$-torus is displaceable by a symplectomorphism, but not by a ...

**1**

vote

**1**answer

275 views

### A question about complex polarization

Let $M$ be a symplectic manifold, Then a subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
$P$ is Lagrangian
P involutive
dim$P\cap\bar P ...

**2**

votes

**1**answer

189 views

### almost holomorphic line bundles

Let $(L,\omega_L) \to (M,\omega_M)$ be a symplectic line bundle (symplectic line=real dimension 2) over a symplectic manifold $M$. Each of these objects can be equipped with an almost complex ...

**1**

vote

**1**answer

157 views

### The set of leaves of the distribution $D$ on coadjoint orbit $O_{\mu}$

Let $G$ be a compact connected Lie group and $O_{\mu}$ be a coadjoint orbit where $\mu\in \mathfrak{g}^*$ and $\mathfrak{g}^*$ is the dual of the Lie algebra of $\mathfrak{g}=\mathrm{Lie}(G)$. Let ...

**2**

votes

**2**answers

180 views

### Does $(M\times M, \omega\times -\omega)$ admit a Lagrangian fibration?

Let $(M,\omega)$ be a manifold endowed with symplectic form. Then the product manifold $M\times M$ with symplectic form $\omega\times -\omega$ is symplectic, and the diagonal submanifold ...

**3**

votes

**2**answers

183 views

### Canonical n plane bundle over Lagrangian Grassmanian

Recall that the Lagrangian Grassmanian, which is denoted by $\Lambda(n)$, is the subset of the standard Grassmanian $G(n,2n)$, which consists lagrangian sub vector spaces of $\mathbb{R}^{2n}$. Lets ...

**5**

votes

**2**answers

320 views

### Why non-compact Calabi-Yau surfaces are not self-mirror?

By the work of Gross and Bernard-Matessi, in dimension 3 $T$-duality should be understood as an exchange of positive and negative local model of Lagrangian torus fibrations, at least in its ...

**3**

votes

**1**answer

132 views

### A question about solutions to Floer's equation which are asymptotic to a stationary point

Let $M$ be a compact symplectic manifold and $H$ a time independent Hamiltonian on $M$. Let $\alpha$ be a solution to Floer's equation
$$ u(t,s): S^1 \times \mathbb{R} \to M$$
$$(du+X_H\otimes ...

**8**

votes

**1**answer

192 views

### Detecting Monodromy in Integrable Systems

Suppose I have a completely integrable system on a symplectic manifold $(M^{2n},\omega)$ with momentum map $H:M \rightarrow \mathbb{R}^n$ that has compact, connected fibers. Further, suppose I know ...

**1**

vote

**1**answer

175 views

### Question about transversality for PSS map in Hamiltonian Floer cohomology

Let X be a compact symplectic manifold and $H_t,J_t$ a Floer regular pair of $\mathbb{S}^1$ dependent Hamiltonians and complex structures. The PSS maps are defined by considering $\mathbb{C}$ with a ...

**2**

votes

**1**answer

229 views

### How to compute Conley-Zehnder indices on prequantization spaces?

My question is pretty much as in the title. On page 100 of his thesis, Bourgeois gives a computation of the CZ(or I suppose more correctly this is the Robbin-Salamon index) index of the Reeb orbits ...

**5**

votes

**0**answers

134 views

### Unobstructed Lagrangian tori

Let $X$ be a compact symplectic manifold, $U$ a Darboux chart, and $L$ a standard Lagrangian torus in $U$. Is $H^1(L)$ weakly unobstructed in the sense of Fukaya-Oh-Ohta-Ono (that is, does $b \in ...

**12**

votes

**3**answers

1k views

### Is there a physical intuition for Darboux's theorem?

We know that there is a physical interpretation for symplectic manifolds (briefly, the fact that a sympletic form assigns to any Hamiltoninan a vector field which describes the motion of particles). ...

**1**

vote

**1**answer

99 views

### wavefront is a coisotropic

I'm thinking about the following example: Let $u_k:=e^{i(kx_1+k^2x_2)}$, where $k\in\mathbb{N}$ and $x_1,x_2\in\mathbb{R}$. Then for the sequence $h_k:=1/(k\sqrt{1+k^2})$ ($h_k\rightarrow0$ as ...

**0**

votes

**1**answer

136 views

### model compact coisotropic submanifold

$\{x_1,\ldots,x_n,\xi_{m+1},\ldots,\xi_n\in\mathbb{R},\xi_1=\ldots=\xi_m=0\}$ is a "model" codimension $m$ coisotropic submanifold of $T^*\mathbb{R}^n$, and is of course noncompact. Is there such a ...

**7**

votes

**0**answers

218 views

### Question on Ionel and Parker's paper: Relative Gromov Witten Invariants

In defining Gromov-Witten invariants using symplectic geometry, most of the trouble is to achieve transversality for moduli spaces of pseudo-holomorphic curves which are multiple covers of simple ...

**5**

votes

**2**answers

273 views

### computing second cohomology $H^2(O_a,\mathbb{Z})\cong \mathbb{Z}^n$ of a generic coadjoint orbit

Let $G$ be a compact , connected and simply connected Lie group and $a\in\mathfrak{g}^*$ (dual of Lie algebra of Lie group $G$). Then let $O_a$ be a generic coadjoint orbit then can we say ...

**2**

votes

**2**answers

774 views

### Kenji Fukaya's Lecture series at Simons center

In the past decade, theory of Kuranishi structures on moduli space of pseudo-holomorphic curves has been in the center of debates between some mathematicians in the field of symplectic geometry.
...

**1**

vote

**2**answers

201 views

### coisotropic submanifolds

I'm thinking about coisotropic/involutive submanifolds of the symplectic phase space $T^*\mathbb{R}^n$ (with coordinates $x_1,\ldots,x_n,\xi_1,\ldots,\xi_n$). As I understand, the smallest ...

**1**

vote

**0**answers

202 views

### quantization for integral coadjoint orbits

I am looking for a referrence for proof of following statement.
Let $G$ be a semi-simple Lie group and $\mathfrak{g}$ be its Lie algebra and $\alpha_k$ be simple roots of $\mathfrak{g}$ and $\mu_0$ ...

**1**

vote

**1**answer

150 views

### Each coadjoint orbit of a compact connected Lie group $G$ admits a $G$-invariant generalized complex structure

I am looking for a proof or counterexample for following assertion
Each coadjoint orbit of a compact connected Lie group $G$ admits
a $G$-invariant generalized complex structure (In sense of ...

**9**

votes

**1**answer

304 views

### An algebraic Hamiltonian vector field with a finite number of periodic orbits

Edit: There is an interesting complete answer for the second part(see the answer by Thomas Kragh). I search for an answer for the first part.
1.Is there a polynomial Hamiltonian ...

**4**

votes

**0**answers

125 views

### Classification of Compact Symplectic Homogeneous Spaces

Let $M=G/H$ be a compact homogeneous space, $G$ a compact Lie Group and $H$ a closed subgroup. Is there some classification, akin to the Kaehler case, for which such manifolds admit a symplectic ...

**3**

votes

**0**answers

127 views

### A symplectic version of critical points

According to the interesting comment of Mohammad F Tehrani, I revise the question as follows:
Assume $n>2$. For what type of compact n dimensional manifolds $M$ we can say:
For every smooth ...

**3**

votes

**1**answer

180 views

### Local holomorphic equations for symplectic divisors

If $(X,\omega)$ is a symplectic manifold and $J\colon TX \to TX$ is an almost complex structure, we know that $J$ is actually a complex structure if and only if the Nijenhuis tensor $N_J(\cdot,\cdot)$ ...

**6**

votes

**1**answer

180 views

### A lagrangian version of the Withney theorem

Let $M$ be a smooth n dimensional manifold. Is there an smooth embedding $f:M \to \mathbb{R}^{2n}$ which image is a Lagrangian submanifold of $\mathbb{R}^{2n}$?

**3**

votes

**1**answer

229 views

### Explicit form for hermitian structure $h$ with respect to $\omega$

Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a complex line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on ...

**1**

vote

**0**answers

137 views

### Explicit formula for hermitian form on coadjoint orbit of $G$ on $\mathfrak{g}^*$

Let $G$ be a compact Lie group and $\mathfrak{g}$ be its Lie algebra and $\mathfrak{g}^*$ be its dual , then I am looking for explicit formula for hermitian form on coadjoint orbit of $G$ on ...

**2**

votes

**0**answers

295 views

### A symplectic structure for cotangent bundle

Before that I mention my question explicitly, I start with my motivation:
Look at $\mathbb{D} \times \mathbb{C}=\{(x_{1},x_{2},y_{1},y_{2})\mid x_{1}^{2}+x_{2}^{2}< 1\}$.This can be identified ...